Henry Ernest Dudeney/Modern Puzzles/138 - The Four Draughtsmen/Solution

Modern Puzzles by Henry Ernest Dudeney: $138$

The four draughtsmen are shown exactly as they stood on a square chequered board --

not necessarily eight squares by eight --

but the ink with which the board was drawn was evanescent,

so that all the diagram except the men has disappeared.

How many squares were there in the board and how am I to reconstruct it?

I know that each man stood in the middle of a square,

one on the edge of each side of the board and no man in a corner.

Construct $AD$.

Construct $CE$ perpendicular to $AD$ and make $CE = AD$.

Because $AD$ stretches from one side of the board to its opposite, joining the centres of two squares, then the same applies to $CE$.

So $E$ is the centre of a square on the same row as $B$.

So join $EB$ and produce it in both directions.

Then:

construct $FG$ through $C$ parallel to $EB$

construct $FH$ through $A$ perpendicular to $EB$ and $FG$

construct $DG$ through $D$ parallel to $FH$.

As $H$ is the centre of a square on a board, we can mark $HE$ all round the square $HFG$ and discover the board is $10 \times 10$.

It remains to construct the actual board itself.